There's already lots of tutorials on what parts you need and sample designs you can recycle for building your own quad, so I won't cover that stuff. Here's just some practical trial-and-error type tips most of those blogs won't tell you:
You're likely to run into different connectors. Deans T, XT-60, 4mm HXT bullet all are common in 3S batteries. You may want to order a few with your parts - also consider that you'll need matching connectors in your quad, the battery charger and the batteries. There's different adapters available, but in the long run the sane approach is to standardize into a single connector. This requires changing the connectors on your batteries as well. XT-60 is a popular choice as it's easy to connect, safe, and easy to solder.
For building up your electric system, and especially connecting your motors on small or medium quads, 3.5mm bullets are handy. Get a version with plastic protective sheaths. Often when you're working on your quad, you're supposed to remove propellers, but lazy people can just remove one of the bullet connectors and place a towel over the propellers. JST connectors or even 2mm bullets work nicely for building LEDs or other 2ndary electric systems for your quad.
Between your battery and the ESCs, you need to build a power distribution. A power distribution board may be handy if it fits into your frame (some frames have one integrated) but you may just solder the wires and connectors together. Placing some connectors into the system may be a good idea for beginners and people who want to modify or extend their system later on. HobbyKing has preassembled power breakout cables with connectors that make building power distribution a breeze. Consider that in addition to each ESC, you may want 1-2 additional wires for powering LEDs, measuring battery voltage, etc.
You should get a few extra connectors - it's likely you will lose one or want to build something you didn't plan originally. Also don't cut your wires too short when building, it's a pain to assemble when it's very tight with the length of the wires, while a few cm of loose wire is never a big problem! Do a google search and check you get heavy enough wire gauge for your current (AWG - smaller number is bigger wire), 12 or 14 AWG may be suitable for your main cables from the battery, lighter gauge between ESCs and motors. Get at least 2 different colours (red for positive and black for negative). Get shrink tube for protecting your connections.
You will soon be crashing your brand new quadcopter once you get it flying. Don't cheap out on the frame - 20 or 30 dollars can get you a decent, robust one. Go for something popular that has documented crash reports. Consider the availability and pricing of spare parts, as well as the ease of repair. Arms, motor mounts and motors take damage in crashes, propellers need to be replaced. Metal has to be bent back in shape, glued plywood parts will rip apart, carbon fiber may crack and needs to be replaced.
You will need at least one LiPo battery - just one is enough to figure out what kind of flight times you get. Just one is not enough for flying though, you should eventually get at least 2-3, switching to another battery between flights is easy. If your quad isn't very heavy with overspecced parts, you don't need a very large battery! What you should be interested in is the C rating of your batteries, ie. how many amps they will be able to put out in practice. Any battery will have some voltage sag under load, and a weak battery can lead to various issues. There is few hard and fast rules for battery sizing, it all depends on how you fly and with what. Cheap batteries cannot be trusted to their C ratings, but as a rule of thumb, if your continuous C rating is higher than the total max rating for your engines, you are probably fine most of the time.
If you plan to put your battery or gopro under the frame, you will need a decent landing gear with some ground clearance. If the battery sits low to the ground, you can just place some foam padding under it.
Also don't cheap out on the charger, a good one will last you long and get your hobby going, also, it may have a resale value. It's best to get one that has a few features, like adjusting the charging current, balancing, a display that shows you what's actually happening...
The cheapest plastic propellers will work fine in the beginning, unless you have a very heavy quad. Get more expensive ones when you have learned to fly and/or know which size of propeller actually works best for you. You may want to try 2 different sizes 1 inch apart. Check that you get prop adapters that fit your motor shafts (many propellers already come with decent adapters).
If you ever plan to fly in the dark or twilight, you will need LEDs. Otherwise you are certain to lose orientation and crash. Figure out what voltages you need for your LED system and build your power accordingly. You don't need to build the actual LEDs right away.
Youtube can be a valuable resource when you are trying to figure out how something is put together, what kind of connectors you need for your ESC (often: just take your pick), how to solder wires to bullet connectors, and so on.
Happy building!
